Finite Group Databasebeta

← C67×#18960 #30698211

Subgroups — C67×#18960 #30698211

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3555C211yesmaximal, characteristic, complemented
2C67×#3217C211yesmaximal, characteristic
3C67×#3208C211yesmaximal, characteristic, complemented
4C67×#3208C211yesmaximal, characteristic, complemented
5C67×#1628C212yesmaximal, complemented
6C67×#1626C212yesmaximal, complemented
7C67×#1623C212yesmaximal, complemented
8C67×#1622C212yesmaximal, complemented
9C67×#1554C211yesmaximal, characteristic, complemented
10C67×#1497C211yesmaximal, characteristic, complemented
11C67×#1482C211yesmaximal, characteristic, complemented
12#563×C67—22no—
13Q82×C67C2211yescharacteristic, complemented
14#557×C67—21no—
15#557×C67C2211yescharacteristic, complemented
16#556×C67—21no—
17#556×C67—21no—
18#556×C67C2211yescharacteristic, complemented
19#554×C67C2211yescharacteristic, complemented
20#552×C67—21no—
21#544×C67—21no—
22#542×C67—21no—
23#541×C67—21no—
24#541×C67—21no—
25#541×C67—22no—
26#533×C67C2211yescharacteristic
27#533×C67C2211yescharacteristic
28C2×#127×C67C2211yescharacteristic
29#529×C67C2211yescharacteristic
30#496×C67—21no—
31#492×C67—21no—
32#492×C67C2212yes—
33#490×C67—21no—
34#486×C67—21no—
35#486×C67C2212yes—
36#460×C67—22no—
37#460×C67C2212yes—
38#460×C67C2212yes—
39#460×C67C2212yes—
40#459×C67C2212yes—
41#450×C67—22no—
42#450×C67—21no—
43#450×C67—21no—
44#450×C67—21no—
45#447×C67—21no—
46#432×C67C2211yescharacteristic
47#431×C67C2211yescharacteristic
48#431×C67C2211yescharacteristic
49#333×C67C2212yescomplemented
50#332×C67C2212yescomplemented
51Q8⋊C8×C67C2212yescomplemented
52Q8⋊C8×C67C2212yescomplemented
53Q8⋊C8×C67C2212yescomplemented
54D4⋊C8×C67C2212yescomplemented
55C22×D4×C67—21no—
56C22×D4×C67—21no—
57C22×D4×C67—21no—
58C2×SD16×C67—24no—
59C2×SD16×C67—22no—
60C2×SD16×C67—22no—
61C2×SD16×C67—22no—
62C2×D8×C67—22no—
63C4⋊Q8×C67—22no—
64C4⋊Q8×C67—21no—
65C4⋊Q8×C67—22no—
66C4⋊Q8×C67—21no—
67C4⋊Q8×C67—21no—
68C4⋊Q8×C67D411yescharacteristic
69C4⋊Q8×C67D411yescharacteristic
70C4⋊Q8×C67C2311yescharacteristic
71C4⋊Q8×C67C2311yescharacteristic
72#127×C67—21no—
73#127×C67—21no—
74#127×C67—21no—
75#127×C67C2312yes—
76#126×C67—22no—
77#126×C67—22no—
78#126×C67—22no—
79#126×C67—22no—
80#126×C67—22no—
81#126×C67—22no—
82#126×C67—21no—
83#125×C67—22no—
84#125×C67—22no—
85#125×C67—22no—
86#125×C67—22no—
87#125×C67—21no—
88#125×C67—22no—
89#125×C67—21no—
90#125×C67—21no—
91#125×C67—21no—
92#125×C67D411yescharacteristic
93#124×C67—22no—
94#124×C67—21no—
95#124×C67D411yescharacteristic
96#123×C67—22no—
97#123×C67—21no—
98C22⋊Q8×C67—22no—
99C22⋊Q8×C67—22no—
100C22⋊Q8×C67—21no—
101C22⋊Q8×C67—21no—
102C22⋊Q8×C67—22no—
103C22⋊Q8×C67—21no—
104C22⋊Q8×C67—21no—
105C22⋊Q8×C67D411yescharacteristic
106C22⋊Q8×C67D411yescharacteristic
107C22⋊Q8×C67D411yescharacteristic
108C22⋊Q8×C67D412yes—
109#121×C67D411yescharacteristic
110C4×Q8×C67—22no—
111C4×Q8×C67—21no—
112C4×Q8×C67—21no—
113C4×Q8×C67—21no—
114C4×Q8×C67—21no—
115C4×Q8×C67—21no—
116C4×Q8×C67—21no—
117C4×Q8×C67—21no—
118C4×Q8×C67D412yes—
119C4×Q8×C67C2311yescharacteristic
120C4×Q8×C67C2311yescharacteristic
121C4×Q8×C67C2311yescharacteristic
122C4×D4×C67—21no—
123C4×D4×C67—21no—
124C4×D4×C67C2311yescharacteristic
125#117×C67—22no—
126#117×C67—21no—
127#117×C67—21no—
128#117×C67—21no—
129#117×C67—21no—
130C2×C4⋊C4×C67—21no—
131C2×C42×C67C2311yescharacteristic
132C4⋊C8×C67—21no—
133C4⋊C8×C67C2312yes—
134C4⋊C8×C67C2312yes—
135#102×C67—22no—
136#102×C67—24no—
137#102×C67—22no—
138#102×C67—22no—
139#102×C67—22no—
140C22⋊C8×C67—21no—
141C22⋊C8×C67—21no—
142C22⋊C8×C67—21no—
143#97×C67—21no—
144C4×C8×C67—21no—
145C4×C8×C67C2312yes—
146C24×C67—41no—
147C2×Q8×C67—22no—
148C2×Q8×C67—21no—
149C2×Q8×C67—21no—
150C2×Q8×C67C2×D411yescharacteristic
151C2×Q8×C67C2×D411yescharacteristic
152C2×Q8×C67C2×D411yescharacteristic
153C2×Q8×C67C2×D412yes—
154C2×D4×C67—21no—
155C2×D4×C67—22no—
156C2×D4×C67—22no—
157C2×D4×C67—22no—
158C2×D4×C67—41no—
159C2×D4×C67—41no—
160C2×D4×C67—41no—
161C2×D4×C67—82no—
162C2×D4×C67—82no—
163C2×D4×C67—82no—
164C2×D4×C67C2×D411yescharacteristic
165C22×C4×C67—22no—
166C22×C4×C67—21no—
167C22×C4×C67C2×D411yescharacteristic
168C22×C4×C67C2×D411yescharacteristic
169C22×C4×C67C2×D411yescharacteristic
170SD16×C67—82no—
171SD16×C67—82no—
172SD16×C67—82no—
173SD16×C67—84no—
174D8×C67—82no—
175C2×C8×C67—22no—
176C2×C8×C67—22no—
177C2×C8×C67—22no—
178C4⋊C4×C67—22no—
179C4⋊C4×C67—22no—
180C4⋊C4×C67—22no—
181C4⋊C4×C67—22no—
182C4⋊C4×C67—22no—
183C4⋊C4×C67—22no—
184C4⋊C4×C67—22no—
185C4⋊C4×C67—22no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—22no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—22no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67—21no—
198C4⋊C4×C67—21no—
199C4⋊C4×C67—21no—
200C4⋊C4×C67—21no—
201C4⋊C4×C67—21no—
202C4⋊C4×C67C2×D411yescharacteristic
203C4⋊C4×C67C2×D411yescharacteristic
204C4⋊C4×C67C2×D411yescharacteristic
205C4⋊C4×C67C2×D411yescharacteristic
206C4⋊C4×C67C2×D412yes—
207C22⋊C4×C67—22no—
208C22⋊C4×C67—22no—
209C22⋊C4×C67—22no—
210C22⋊C4×C67—21no—
211C22⋊C4×C67—22no—
212C22⋊C4×C67—21no—
213C22⋊C4×C67—21no—
214C22⋊C4×C67—21no—
215C22⋊C4×C67—21no—
216C22⋊C4×C67—21no—
217C22⋊C4×C67—21no—
218C42×C67—22no—
219C42×C67—22no—
220C42×C67—22no—
221C42×C67—22no—
222C42×C67—21no—
223C42×C67—21no—
224C42×C67—21no—
225C42×C67C2×D411yescharacteristic
226C42×C67C2×D411yescharacteristic
227C42×C67C2×D411yescharacteristic
228C42×C67C2411yescharacteristic
229C23×C67—21no—
230C23×C67—42no—
231C23×C67—82no—
232C23×C67—82no—
233C23×C67—82no—
234C23×C67#12011yescharacteristic
235Q8×C67—22no—
236Q8×C67—22no—
237Q8×C67—22no—
238Q8×C67—42no—
239Q8×C67—42no—
240Q8×C67—81no—
241Q8×C67—82no—
242D4×C67—22no—
243D4×C67—44no—
244D4×C67—42no—
245D4×C67—42no—
246D4×C67—42no—
247D4×C67—42no—
248D4×C67—82no—
249D4×C67—82no—
250D4×C67—82no—
251D4×C67—81no—
252C2×C4×C67—22no—
253C2×C4×C67—22no—
254C2×C4×C67—21no—
255C2×C4×C67—21no—
256C2×C4×C67—21no—
257C2×C4×C67—22no—
258C2×C4×C67—21no—
259C2×C4×C67—21no—
260C2×C4×C67—21no—
261C2×C4×C67—22no—
262C2×C4×C67—21no—
263C2×C4×C67—41no—
264C2×C4×C67—42no—
265C2×C4×C67—82no—
266C2×C4×C67#12011yescharacteristic
267C2×C4×C67#12011yescharacteristic
268C2×C4×C67#12011yescharacteristic
269C2×C4×C67C22×D411yescharacteristic
270C2×C4×C67C22×D411yescharacteristic
271C2×C4×C67C22×D411yescharacteristic
272C8×C67—42no—
273C8×C67—42no—
274C8×C67—42no—
275C22×C67—21no—
276C22×C67—21no—
277C22×C67—21no—
278C22×C67—41no—
279C22×C67—41no—
280C22×C67—41no—
281C22×C67—82no—
282C22×C67—82no—
283C22×C67—82no—
284C22×C67—164no—
285C22×C67C2×#12011yescharacteristic
286C4×C67—21no—
287C4×C67—21no—
288C4×C67—21no—
289C4×C67—21no—
290C4×C67—41no—
291C4×C67—41no—
292C4×C67—41no—
293C4×C67—41no—
294C4×C67—42no—
295C4×C67—41no—
296C4×C67—42no—
297C4×C67—41no—
298C4×C67—41no—
299C4×C67#57712yes—
300C4×C67#57712yes—
301#18960C6711yesmaximal, characteristic, direct product factor, Sylow
302C2×C67—41no—
303C2×C67—81no—
304C2×C67—162no—
305C2×C67#301011yescharacteristic
306C2×C67#301011yescharacteristic
307C2×C67#302111yescharacteristic
308#3555C2×C6711yescharacteristic, complemented
309#3217C2×C6711yescharacteristic
310#3208C2×C6711yescharacteristic, complemented
311#3208C2×C6711yescharacteristic, complemented
312#1628C2×C6712yescomplemented
313#1626C2×C6712yescomplemented
314#1623C2×C6712yescomplemented
315#1622C2×C6712yescomplemented
316#1554C2×C6711yescharacteristic, complemented
317#1497C2×C6711yescharacteristic, complemented
318#1482C2×C6711yescharacteristic, complemented
319C67#1896011yescharacteristic, direct product factor, Sylow
320#563—22no—
321Q82C22×C6711yescharacteristic, complemented
322#557—21no—
323#557C22×C6711yescharacteristic, complemented
324#556—21no—
325#556—21no—
326#556C22×C6711yescharacteristic, complemented
327#554C22×C6711yescharacteristic, complemented
328#552—21no—
329#544—21no—
330#542—21no—
331#541—21no—
332#541—21no—
333#541—22no—
334#533C22×C6711yescharacteristic
335#533C22×C6711yescharacteristic
336C2×#127C22×C6711yescharacteristic
337#529C22×C6711yescharacteristic
338#496—21no—
339#492—21no—
340#492C22×C6712yes—
341#490—21no—
342#486—21no—
343#486C22×C6712yes—
344#460—22no—
345#460C22×C6712yes—
346#460C22×C6712yes—
347#460C22×C6712yes—
348#459C22×C6712yes—
349#450—22no—
350#450—21no—
351#450—21no—
352#450—21no—
353#447—21no—
354#432C22×C6711yescharacteristic
355#431C22×C6711yescharacteristic
356#431C22×C6711yescharacteristic
357#333C22×C6712yescomplemented
358#332C22×C6712yescomplemented
359Q8⋊C8C22×C6712yescomplemented
360Q8⋊C8C22×C6712yescomplemented
361Q8⋊C8C22×C6712yescomplemented
362D4⋊C8C22×C6712yescomplemented
363C22×D4—21no—
364C22×D4—21no—
365C22×D4—21no—
366C2×SD16—24no—
367C2×SD16—22no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×D8—22no—
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—22no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8D4×C6711yescharacteristic
377C4⋊Q8D4×C6711yescharacteristic
378C4⋊Q8C23×C6711yescharacteristic
379C4⋊Q8C23×C6711yescharacteristic
380#127—21no—
381#127—21no—
382#127—21no—
383#127C23×C6712yes—
384#126—22no—
385#126—22no—
386#126—22no—
387#126—22no—
388#126—22no—
389#126—22no—
390#126—21no—
391#125—22no—
392#125—22no—
393#125—22no—
394#125—22no—
395#125—21no—
396#125—22no—
397#125—21no—
398#125—21no—
399#125—21no—
400#125D4×C6711yescharacteristic
401#124—22no—
402#124—21no—
403#124D4×C6711yescharacteristic
404#123—22no—
405#123—21no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—21no—
409C22⋊Q8—21no—
410C22⋊Q8—22no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8D4×C6711yescharacteristic
414C22⋊Q8D4×C6711yescharacteristic
415C22⋊Q8D4×C6711yescharacteristic
416C22⋊Q8D4×C6712yes—
417#121D4×C6711yescharacteristic
418C4×Q8—22no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8D4×C6712yes—
427C4×Q8C23×C6711yescharacteristic
428C4×Q8C23×C6711yescharacteristic
429C4×Q8C23×C6711yescharacteristic
430C4×D4—21no—
431C4×D4—21no—
432C4×D4C23×C6711yescharacteristic
433#117—22no—
434#117—21no—
435#117—21no—
436#117—21no—
437#117—21no—
438C2×C4⋊C4—21no—
439C2×C42C23×C6711yescharacteristic
440C4⋊C8—21no—
441C4⋊C8C23×C6712yes—
442C4⋊C8C23×C6712yes—
443#102—22no—
444#102—22no—
445#102—24no—
446#102—22no—
447#102—22no—
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—21no—
451#97—21no—
452C4×C8—21no—
453C4×C8C23×C6712yes—
454C24—41no—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8C2×D4×C6711yescharacteristic
459C2×Q8C2×D4×C6711yescharacteristic
460C2×Q8C2×D4×C6711yescharacteristic
461C2×Q8C2×D4×C6712yes—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—82no—
470C2×D4—82no—
471C2×D4—82no—
472C2×D4C2×D4×C6711yescharacteristic
473C22×C4—22no—
474C22×C4—21no—
475C22×C4C2×D4×C6711yescharacteristic
476C22×C4C2×D4×C6711yescharacteristic
477C22×C4C2×D4×C6711yescharacteristic
478SD16—82no—
479SD16—82no—
480SD16—82no—
481SD16—84no—
482D8—82no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4C2×D4×C6711yescharacteristic
511C4⋊C4C2×D4×C6711yescharacteristic
512C4⋊C4C2×D4×C6711yescharacteristic
513C4⋊C4C2×D4×C6711yescharacteristic
514C4⋊C4C2×D4×C6712yes—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42C2×D4×C6711yescharacteristic
534C42C2×D4×C6711yescharacteristic
535C42C2×D4×C6711yescharacteristic
536C42C24×C6711yescharacteristic
537C23—21no—
538C23—42no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23#120×C6711yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—22no—
546Q8—42no—
547Q8—42no—
548Q8—81no—
549Q8—82no—
550D4—22no—
551D4—44no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—82no—
557D4—82no—
558D4—82no—
559D4—81no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4#120×C6711yescharacteristic
575C2×C4#120×C6711yescharacteristic
576C2×C4#120×C6711yescharacteristic
577C2×C4C22×D4×C6711yescharacteristic
578C2×C4C22×D4×C6711yescharacteristic
579C2×C4C22×D4×C6711yescharacteristic
580C8—42no—
581C8—42no—
582C8—42no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—164no—
593C22C2×#120×C6711yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4#577×C6712yes—
608C4#577×C6712yes—
609C2—41no—
610C2—81no—
611C2—162no—
612C2C67×#301011yescharacteristic
613C2C67×#301011yescharacteristic
614C2C67×#302111yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3217C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C67×#3555C2semidirect
3. C67×#3208C2semidirect
4. C67×#3208C2semidirect
5. C67×#1628C2semidirect
6. C67×#1626C2semidirect
7. C67×#1623C2semidirect
8. C67×#1622C2semidirect
9. C67×#1554C2semidirect
10. C67×#1497C2semidirect
11. C67×#1482C2semidirect
13. Q82×C67C22semidirect
15. #557×C67C22semidirect
18. #556×C67C22semidirect
19. #554×C67C22semidirect
49. #333×C67C22semidirect
50. #332×C67C22semidirect
51. Q8⋊C8×C67C22semidirect
52. Q8⋊C8×C67C22semidirect
53. Q8⋊C8×C67C22semidirect
54. D4⋊C8×C67C22semidirect
301. #18960C67direct (N × H)
308. #3555C2×C67semidirect
310. #3208C2×C67semidirect
311. #3208C2×C67semidirect
312. #1628C2×C67semidirect
313. #1626C2×C67semidirect
314. #1623C2×C67semidirect
315. #1622C2×C67semidirect
316. #1554C2×C67semidirect
317. #1497C2×C67semidirect
318. #1482C2×C67semidirect
319. C67#18960direct (N × H)
321. Q82C22×C67semidirect
323. #557C22×C67semidirect
326. #556C22×C67semidirect
327. #554C22×C67semidirect
357. #333C22×C67semidirect
358. #332C22×C67semidirect
359. Q8⋊C8C22×C67semidirect
360. Q8⋊C8C22×C67semidirect
361. Q8⋊C8C22×C67semidirect
362. D4⋊C8C22×C67semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C67×#35552
C67×#32172
C67×#32082
C67×#16282
C67×#16262
C67×#16232
C67×#16222
C67×#15542
C67×#14972
C67×#14822
#1896067